3Extension of AF4.1 Solve two-step linear equations and inequalities in one variable over the rational numbers, interpret the solution or solutions in the context from which they arose, and verify the reasonableness of the results.CaliforniaStandards
4A multi-step equation requires more than two steps to solve A multi-step equation requires more than two steps to solve. To solve a multi-step equation, you may have to simplify the equation first by combining like terms.
5Additional Example 1: Solving Equations That Contain Like Terms Solve.8x x – 2 = 37Commutative Property of Addition8x + 3x + 6 – 2 = 3711x + 4 = 37 Combine like terms.– 4 – 4 Since 4 is added to 11x, subtract 4 from both sides.11x = 33331111x=Since x is multiplied by 11, divide both sides by 11.x = 3
6Check It Out! Example 1Solve.9x x – 2 = 42Commutative Property of Addition9x + 4x + 5 – 2 = 4213x + 3 = 42 Combine like terms.– 3 – 3 Since 3 is added to 13x, subtract 3 from both sides.13x = 39391313x=Since x is multiplied by 13, divide both sides by 13.x = 3
7If an equation contains fractions, it may help to multiply both sides of the equation by the least common denominator (LCD) to clear the fractions before you isolate the variable.
8Additional Example 2A: Solving Equations That Contain Fractions Solve.+ = –5n4743474–35n= 4( ) ( )Multiply both sides by 4.( ) ( ) ( )5n47–3= 4Distributive Property( ) ( ) ( )5n47–3= 4Simplify.5n + 7 = –3
9Additional Example 2A Continued – 7 –7 Since 7 is added to 5n, subtract from both sides.5n = –105n5–10=Since n is multiplied by 5, divide both sides by 5n = –2
10The least common denominator (LCD) is the smallest number that each of the denominators will divide into evenly.Remember!
11Additional Example 2B: Solving Equations That Contain Fractions Solve.+ – =x27x91723( ) ( )x237x917– = 18Multiply both sides by 18, the LCD.18( ) + 18( ) – 18( ) = 18( )7x9x2173Distributive Property18( ) + 18( ) – 18( ) = 18( )7x9x21732926Simplify.111114x + 9x – 34 = 12
12Additional Example 2B Continued 23x – 34 = Combine like terms.Since 34 is subtracted from x, add 34 to both sides.23x = 46=23x2346Since x is multiplied by 23, divide t both sides by 23.x = 2
13Distributive Property Check It Out! Example 2ASolve.+ = –3n4541454–13n= 4( ) ( )Multiply both sides by 4.( ) ( ) ( )3n45–1= 4Distributive Property( ) ( ) ( )3n45–1= 4111Simplify.1113n + 5 = –1
14Check It Out! Example 2A Continued – 5 – Since 5 is added to 3n, subtract 5 from both sides.3n = –63n3–6=Since n is multiplied by 3, divideboth sides by 3.n = –2
15Multiply both sides by 9, the LCD. x 3 1 5x 9 13 ( ) ( ) 9 + – = 9 Check It Out! Example 2BSolve.+ – =x35x91313Multiply both sides by 9, the LCD.x315x913( ) ( )– = 99( ) + 9( ) – 9( ) = 9( )5x9x3131Distributive Property9( ) + 9( ) – 9( ) = 9( )5x9x31311313Simplify.11115x + 3x – 13 = 3
16Check It Out! Example 2B Continued 8x – 13 = Combine like terms.Since 13 is subtracted from 8x, add 13 to both sides.8x = 16=8x816Since x is multiplied by 8, divide t both sides by 8.x = 2
17Additional Example 3: Travel Application On Monday, David rides his bicycle m miles in 2 hours. On Tuesday, he rides three times as far in 5 hours. If his average speed for the two days is 12 mi/h, how far did he ride on Monday? Round your answer to the nearest tenth of a mile.David’s average speed is his total distance for the two days divided by the total time.Total distanceTotal time=average speed
18Additional Example 3 Continued 2 + 5= 12m + 3mSubstitute m + 3m for total distance and for total time.7= 124mSimplify.= 7(12)74mMultiply both sides by 7.4m = 8484 44m 4=Divide both sides by 4.m = 21David rode 21.0 miles.
19Check It Out! Example 3On Saturday, Penelope rode her scooter m miles in 3 hours. On Sunday, she rides twice as far in 7 hours. If her average speed for two days is 20 mi/h, how far did she ride on Saturday? Round your answer to the nearest tenth of a mile.Penelope’s average speed is her total distance for the two days divided by the total time.Total distanceTotal time=average speed
20Check It Out! Example 3 Continued 3 + 7= 20m + 2mSubstitute m + 2m for total distance and for total time.10= 203mSimplify.= 10(20)103mMultiply both sides by 10.3m = 2003m 3=Divide both sides by 3.m 66.67Penelope rode approximately 66.7 miles.
21Lesson Quiz Solve. 1. 6x + 3x – x + 9 = 33 2. 29 = 5x + 21 + 3x x = 3 =5. Linda is paid double her normal hourly rate for each hour she works over 40 hours in a week. Last week she worked 52 hours and earned $544. What is her hourly rate?x = 3x = 158x8338x = 28x = 1916– =6x72x212521$8.50